Abstract: If a field ${\boldsymbol A}$ of class ${\mathcal C}^2$ of positive-definite symmetric matrices of order two and a field ${\boldsymbol B}$ of class ${\mathcal C}^1$ of symmetric matrices of order two satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply-connected open subset $\omega$ of ${\mathbb R}^2$, then there exists an immersion ${\boldsymbol \theta}\in {\mathcal C}^3(\omega; {\mathbb R}^3)$, uniquely determined up to proper isometries in ${\mathbb R}^3$, such that ${\boldsymbol A}$ and ${\boldsymbol B}$ are the first and second fundamental forms of the surface ${\boldsymbol \theta}(\omega)$. Let $\dot{\boldsymbol \theta}$ denote the equivalence class of ${\boldsymbol \theta}$ modulo proper isometries in ${\mathbb R}^3$ and let ${\mathcal F}:({\boldsymbol A},{\boldsymbol B}) \to \dot{\boldsymbol \theta}$ denote the mapping determined in this fashion.

The first objective of this paper is to show that, if $\omega$ satisfies a certain ``geodesic property'' (in effect a mild regularity assumption on the boundary of $\omega$) and if the fields ${\boldsymbol A}$ and ${\boldsymbol B}$ and their partial derivatives of order $\leq 2$, resp. $\leq 1$, have continuous extensions to ${\overline\omega}$, the extension of the field ${\boldsymbol A}$ remaining positive-definite on ${\overline\omega}$, then the immersion ${\boldsymbol \theta}$ and its partial derivatives of order $\leq 3$ also have continuous extensions to ${\overline\omega}$.

The second objective is to show that, if $\omega$ satisfies the geodesic property and is bounded, the mapping ${\mathcal F}$ can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces ${\mathcal C}^2({\overline\omega})\times{\mathcal C}^1({\overline\omega})$ for the continuous extensions of the matrix fields $({\boldsymbol A},{\boldsymbol B})$, and ${\mathcal C}^3({\overline\omega})$ for the continuous extensions of the immersions~${\boldsymbol \theta}$.